Chapter 1
Section 1.3 New Functions from Old Functions
Transformations of functions
Translations
𝑦 = 𝑓(𝑥) + 𝑐, shifts the graph of 𝑦 = 𝑓(𝑥) a distance of c units upward
𝑦 = 𝑓(𝑥) − 𝑐, shifts the graph of 𝑦 = 𝑓(𝑥) a distance of c units downward
𝑦 = 𝑓(𝑥 − 𝑐), shifts the graph of 𝑦 = 𝑓(𝑥) a distance of c units to the right
𝑦 = 𝑓(𝑥 + 𝑐), shifts the graph of 𝑦 = 𝑓(𝑥) a distance of c units to the left
Stretching and Reflecting
𝑦 = 𝑐𝑓(𝑥), stretch the graph of 𝑦 = 𝑓(𝑥) vertically by a factor of c
1
𝑦 = 𝑐 𝑓(𝑥), compresses the graph of 𝑦 = 𝑓(𝑥) vertically by a factor of c
𝑦 = 𝑓(𝑐𝑥), compresses the graph of 𝑦 = 𝑓(𝑥) horizontally by a factor of c
𝑥
𝑦 = 𝑐𝑓 (𝑐 ), stretch the graph of 𝑦 = 𝑓(𝑥) horizontally by a factor of c
𝑦 = −𝑓(𝑥), reflects the graph of 𝑦 = 𝑓(𝑥) about the x-axis
𝑦 = 𝑓(−𝑥), reflects the graph of 𝑦 = 𝑓(𝑥) about the y-axis
𝒚 = 𝒂 + 𝒃𝒇(𝒄(𝒙 + 𝒅))
a) Vertical shift factor (shifts down when negative)
b) Vertical stretch factor (reflects on x-axis if negative)
c) Horizontal stretch factor (reflects on y-axis if negative)
d) Horizontal shift factor (shifts to the right when negative)
Amplitude – the factor by which to stretch the sine curve vertically.
point and l is the lowest point on the curve.
1
(ℎ
2
− 𝑙) where h is the highest
Absolute Value – the part of the curve that lies above the x axis stays the same while the part below is
reflected in the x-axis
Combinations of Functions
𝐴 is the domain of 𝑓(𝑥) and 𝐵 is the domain of 𝑔(𝑥)
Addition
Domain: 𝐴 ∩ 𝐵
Subtraction
Domain: 𝐴 ∩ 𝐵
Multiplication
Domain: 𝐴 ∩ 𝐵
Division
Domain: (−∞, 1) ∪ (1, ∞)
Composition
The domain of (𝑓 ∘ 𝑔) is the set of all x in the domain
of g such that 𝑔(𝑥) is the domain of 𝑓
(𝑓 + 𝑔)(𝑥) = 𝑓(𝑥) + 𝑔(𝑥)
(𝑓 − 𝑔)(𝑥) = 𝑓(𝑥) − 𝑔(𝑥)
(𝑓𝑔)(𝑥) = 𝑓(𝑥)𝑔(𝑥)
𝑓
𝑓(𝑥)
( ) (𝑥) =
𝑔
𝑔(𝑥)
(𝑓 ∘ 𝑔)(𝑥) = 𝑓(𝑔(𝑥))
Exponential functions (Has a y intercept of 1)
𝑓(𝑥) = 𝑎 𝑥 where 𝑎 > 0
Certain values of 𝑎 come up a lot
𝑎 = 2, 10, 𝑒
𝑒 = 2.712 …
The slope of the graph of 𝑦 = 𝑒 𝑥 𝑎𝑡 𝑎𝑛𝑦 𝑝𝑜𝑖𝑛𝑡 𝑥 𝑖𝑠 𝑗𝑢𝑠𝑡 𝑒 𝑥 .
Def:
1 𝑛
1) 𝑒 = lim (1 + 𝑛)
2)
𝑛→∞
1
1
1
𝑒 = 1 + 2 + 6 + 24 + ⋯
1
1
1
1
= !1 + !2 + !3 + !4 + ⋯
𝑒 𝑖𝜃 = cos 𝜃 + 𝑖 sin 𝜃
𝑖 2 = −1
Properties of exponents
1. 𝑎0 = 1
1
2. 𝑎−𝑐 = 𝑎𝑐
3. 𝑎𝑏+𝑐 = 𝑎𝑏 𝑎𝑐
𝑐
𝑐
4. (𝑎𝑏 ) = 𝑎𝑏𝑐 ≠ 𝑎𝑏
1
𝑐
5. 𝑎𝑐 = √𝑎
6. 𝑎𝑐 𝑏 𝑐 = (𝑎𝑏)𝑐
Ex:
𝑐
𝑐
( √𝑎) = 𝑎
(√𝑥 − 1√𝑥 + 1)
1
= [(𝑥 − 1)(𝑥 + 1)]2
= √𝑥 2 − 1
Hyperbolic Trigonometric functions
𝑒 𝑥 +𝑒 −𝑥
2
𝑒 𝑥 −𝑒 −𝑥
sinh(𝑥) =
2
𝑒 𝑥 −𝑒 −𝑥
tanh(𝑥) = 𝑒 𝑥 +𝑒 −𝑥
cosh(𝑥) =
Logarithmic functions
log 𝑎 (𝑏) = 𝑐
or
𝑎𝑐 = 𝑏
(𝑎 > 0) & (𝑏 > 0)
Properties of Logarithmic Functions
1) log 𝑎 (𝑏𝑐) = log 𝑎 (𝑏) +log 𝑎 (𝑐)
2) log 𝑎 (1) = 0
3) log 𝑎 (𝑏 𝑐 ) = 𝑐 log 𝑎 (𝑏)
Common 𝑎 = 2, 10, 𝑒
log 𝑒 (𝑥) = ln(𝑥) = “lon x” (natural logarithm)
1
𝑦 = 𝑙𝑜𝑛(𝑥): the slope of it at 𝑥 = 𝑥
Inverse Functions
𝑔(𝑥) is the inverse of 𝑓(𝑥)
- If f(𝑔(𝑥)) = 𝑥
- Usually we denote the inverse of f(x) by 𝑓 −1 (𝑥)
- 𝑓 −1 (𝑓𝑥) = 𝑥 -->
1. The inverse of f(𝑥) = 𝑥
- 𝑓 −1 (𝑥) = 𝑥 = 𝑓(𝑥)
2. The inverse of 𝑔(𝑥) = −𝑥
- 𝑔−1 (𝑥) = −𝑥 = 𝑔(𝑥)
1
3. ℎ(𝑥) = 𝑥 has inverse
-
ℎ−1 (𝑥) =
1
𝑥
= ℎ(𝑥)
4. 𝑓(𝑥) = ln(𝑥) has inverse 𝑓 −1 (𝑥) = 𝑒 𝑥
5. 𝑔(𝑥) = 𝑒 𝑥 has inverse 𝑔−1 (𝑥) = ln(𝑥)
-
In general (𝑓 −1 )−1 (𝑥) = 𝑓(𝑥)
6. The inverse of tan(𝑥) = tan−1(𝑥) or arctan(𝑥)